I will discuss trend filtering, a newly proposed tool of Steidl et al. (2006), Kim et al. (2009) for nonparametric regression. The trend filtering estimate is defined as the minimizer of a penalized least squares criterion, in which the penalty term sums the absolute kth order discrete derivatives over the input points. I will give an overview of some interesting connections between these estimates and adaptive spline estimation, and also of the provable statistical superiority of trend filtering to other common nonparametric regression tools, such as smoothing splines and kernel smoothing.  I will then focus on some extensions of trend filtering beyond the univariate setting, namely, to multivariate data and (separately) to graph-based data.  This represents joint work with Veeranjaneyulu Sadhanala, Yu-Xiang Wang, James Sharpnack, and Alex Smola.